The Number e
āe is the base of 'natural growth'.ā
The formula
e = lim(nāā) (1 + 1/n)āæ ā 2.71828How to read it: the number you approach by compounding 1/n over and over in n steps, about 2.718
- n
- ā how many times you compound (growing without bound)
- (1+1/n)āæ
- ā the value grown by compounding in n steps
- e
- ā the limit it approaches ā 2.71828
The hook
As famous as Ļ but far less understood, e hides almost everywhere nature 'grows'.
In plain words
Take 100% interest, chop it into ever-finer compounding steps, and as the number of steps goes to infinity the result approaches e ā 2.718.
The intuition
One dollar at 100% annual interest. Paid once at year-end, you get $2. Half-and-half twice gives (1.5)Ā²=$2.25. The finer you slice, the more it grows ā but no matter how fine, it never passes 2.71828ā¦ and converges to it.
How it's built
The key part is (1+1/n)āæ. Inside the parentheses is 'grow a little (1+1/n)', the exponent n is 'repeat that n times'. Grow n and these two tug against each other, settling on e.
Example
n=1 ā 2, n=2 ā (1.5)Ā²=2.25, n=12 ā about 2.613, n=ā ā 2.71828ā¦. It keeps rising but never breaks through the ceiling that is e.
Common mistake
e isn't an arbitrary irrational number. It's the one base whose function differentiates to itself, which is why it turns up so specially in nature.
Where it's used
Continuous compounding, population growth, radioactive decay, cooling, probability (Poisson) ā any process that changes in proportion to its own size.
Where it came from
Jacob Bernoulli met this limit in a compound-interest problem, and Euler fixed it as a constant by naming it e.
Prerequisites
Quick check
What is the approximate value of e?
- 1.414
- 2.718ā
- 3.141
- 1.618
Practice
What is (1+1/n)āæ when n=1?
Answer: 2
- (1+1/1)Ā¹ = 2Ā¹
- = 2
Takeaway: Compounding just once gives 2, the starting point toward e.
What is (1+1/2)Ā²? (compounding in two half-steps)
Answer: 2.25
- 1+1/2 = 1.5
- 1.5Ā² = 2.25
Takeaway: Finer steps creep a little closer to 2.718ā¦.
With 100% annual interest compounded every half-year, what does $1 become after a year? Compare with continuous compounding (e).
Answer: undefined
- Two 50% steps: (1.5)Ā² = $2.25
- Compounded infinitely finely it would be e ā $2.718 ā larger
Takeaway: Finer compounding earns more, but the ceiling is a factor of e.